Weakly S-additive measures

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1 Weakly S-additive measures Zuzana Havranová Dept. of Mathematics Slovak University of Technology Radlinského, Bratislava Slovakia Martin Kalina Dept. of Mathematics Slovak University of Technology Radlinského, Bratislava Slovakia Abstract In this paper we study Lukasiewicz filters as special weakly S-additive measures. Further, we introduce T - conditional S-measures and we investigate them for some particularly chosen t-norms T and t-conorms S. Keywords: Lukasiewicz filter, S- additivity, Conditioning. Introduction and preliminaries Filters have been already generalized to fuzzy setting in papers, e.g. [, 2, 3, 4], where socalled generalized filters have been studied. We have investigated this topic in paper [5], introducing so-called Lukasiewicz filters. Filters are in fact special fuzzy measures. In this paper we study the relationship between Lukasiewicz filters and fuzzy measures. The crucial notion will be the weak S-additivity, which is a modification of S-additivity (see, e.g. [6]). In the whole paper we will use the following notation: Z is our universe (it might be either countable or uncountable) In case the universe is necessarily at most countable, it will be denoted as X = x, x 2,...} W = w, w 2,...} is the weighting sequence of the corresponding elements of X S is some left-continuous t-conorm S j (w ij ) = S (w i, w i2,...) Remark Let S be a left-continuous t-conorm and (c i ) i= be a system of constants, each c i [0, ]. Then S n = S(c, c 2,..., c n ) is a nondecreasing sequence with values in [0, ]. Since the t-norm S is left-continuous, there exists lim S n = L. n This limit will be denoted by S i (c i ). Let us recall the basic definitions: Definition Let us have a set Z. Then F : 2 Z [0, ] is said to be a Lukasiewicz filter on Z if and only if the following are satisfied:. F(Z) =, F( ) = 0 2. all A, B Z such that A B, yield 3. all A, B Z yield F(A) F(B) F(A B) T L (F(A), F(B)) We will use just the abbreviation Lukasiewicz filter without specifying the universe Z, since this will be known from the context. Definition 2 F : 2 X [0, ], defined by, if A = X F(A) = S j (w ij ), if A = x i, x i2,...} is said to be a (T L, S)-filter, if and only if F is a Lukasiewicz filter.

2 Definition 3 Let (Z, O) be a measurable space and S some left-continuous t-conorm. A measure ν : O [0, ] is said to be S- additive, if and only if for every couple A, B O of disjoint sets the following holds: ν(a B) = S (ν(a), ν(b)). () If formula () holds just for couples of disjoint sets A, B O such that A B X, then the measure ν will be called weakly S-additive. Immediately by Definition 3 we get the following: Lemma Each (T L, S)-filter is a weakly S- additive measure. However, as the following example shows, not all weakly S-additive measures are Lukasiewicz filters. Example (a) Let S M be maximum. Put all weights w i = 0.5. Then we get the following (T L, S M )-filter: 0, if A = F SM (A) =, if A = X 0.5 otherwise On the other hand, let X = x, x 2 } and w = 0.6 w 2 = 0.5, then there exists the corresponding weakly S M -additive measure µ, but this is not a (T L, S M )-filter, since T L (µ(x }), µ(x 2 })) = 0. i.e., the property 3 of Lukasiewicz filters is violated. (b) Let S P be the probabilistic sum. Let X = x, x 2, x 3, x 4 }. Put all weights w i = 0.3. Then we may construct the following weakly S P -additive measure:, if A = X µ(a) = 0.7 A otherwise where A denotes the cardinality of A. This is not a (T L, S P )-filter, since µ(x, x 2 }) = µ(x 3, x 4 }) = 0.5 i.e., they are disjoint but T L (µ(x, x 2 }), µ(x 3, x 4 })) > 0 hence the property 3 of Lukasiewicz filters is violated. On the other hand, if we put all the weights w i = 0.27, then we may construct the (T L, S P )-filter by:, if A = X F SP (A) = 0.73 A otherwise (c) Let S L be the Lukasiewicz t-conorm. Let us have a weighting sequence W = w i } i. We may construct the (T L, S L )-filter if and only if w i. Particularly, each probability i distribution on X is a (T L, S L )-filter. 2 Construction of weakly S-additive measures Assume that (Z, ρ) is a locally compact metric space. Denote by B 2 Z the Borel σ-algebra. Definition 4 A Lukasiewicz filter F B, restricted to the Borel σ-algebra B, is said to be a Lukasiewicz B-filter. Theorem Let ν : B [0, ] be a weakly S- additive measure with ν(x) = and fulfilling ν(a) + ν(b) Then ν is a Lukasiewicz B-filter. if A and B are disjoint. Fix some weakly S-additive measure ν : B [0, ]. Define µ ν : 2 Z [0, ] by µ ν (A) = supν(b); B B, B A} and µ ν : 2 Z [0, ] by µ ν(a) = infν(b); B B, B A} Then the following holds: Lemma 2 µ ν is a Lukasiewicz filter. If µ ν µ ν, then µ ν is not weakly S-additive. Assume that Z = [0, ] and ρ is the usual Euclidian metric. Let us consider a sequence of finite sets, X n } n, such that X n = 2 n and a sequence of weakly S-additive measures ν n : 2 Xn [0, ], all of them uniformly distributed, such that S (ν i (A), ν i (A)) = ν i (B)

3 where A X i, B X i and A = B. In fact, ν i (x }) = µ([0, 2 i ]) where µ : B [0, ] is the measure we are constructing. Take some interval A [0, ]. expressed as diadic number: ρ(a) = a i 2 i, where a i = i= Its length is 0 Now, we can define the value µ(a) for any interval A [0, ] by if A = X µ(a) = ( ai µ([0, 2 i ]) ) otherwise. S i= 3 Conditioning using weakly S-additive measures The motivation for this chapter is the classical definition of the conditional probability: P (A B) = P (A B) P (B) (2) assuming P (B) 0, where A, B are from a σ- algebra S of P -measurable sets. The formula (2) can be rewritten into the form P (A B) = supx [0, ]; x P (B) P (A B)}. (3) I.e., the conditional probability might be viewed at as a residual operator with respect to the product. Now, we generalize formula (3) in such a way that instead of an additive probability we will use some weakly S- additive measure µ and instead of the product we will use some t-norm. More precisely, let (Z, B, µ) be the Borel space with a weakly S-additive measure µ and T be a t-norm. Denote B 0 = B B; µ(b) 0}. Then we denote by µ T : B B 0 the following two-place function: µ T (A B) = supx [0, ]; T (µ(b), x) µ(a B)} (4) Lemma 3 Let S be some left-continuous t- conorm and (Z, B, µ) be the Borel space with a weakly S-additive measure µ. Then the following are satisfied: (a) Let T be an arbitrary t-norm. Then µ T (A B) = for all B B 0 and all A B such that µ(a B) = µ(b). (b) All B B 0 yield µ T ( B) = 0 if and only if T is a t-norm with no zero divisors. Of course, it is quite natural to ask µ T ( B) = 0 for all B B 0. This is the reason, why we will use only those t-norms T which have no zero divisors. Definition 5 Let (Z, B, µ) be a measurable space with a weakly S-additive measure µ and T be a t-norm with no zero divisors. Then µ T : B B 0, defined by formula (4), is said to be a T -conditional S-measure. In what follows we will fix some Borel space (Z, B). The measure µ will always be defined on Borel sets B. The next theorem is a modification of the well-known full probability theorem. Theorem 2 Assume that S be a leftcontinuous t-conorm. Let us have a weakly S- additive measure µ and a continuous t-norm, T, with no zero divisors. Assume B, B c B 0. Then the following is satisfied: µ(a) = S(T (µ T (A B), µ(a B)), T (µ T (A B c ), µ(a B c ))) Theorem 3 Let T be a t-norm with no zero divisors and S some left-continuous t-conorm. Fix some B B 0 and denote B B = A B; A B}. Assume that µ is weakly S-additive and moreover that it is a Lukasiewicz B-filter. Then the following holds: if S S L, then µ T (. B) is a Lukasiewicz B B - filter.

4 Now, we are going to study the S-additivity of T -conditional S-measures for some particularly chosen t-norms T and t-conorms S. As we have found out by Lemma 3(a), the only kind of S-additivity, which is reasonable to consider, is the following: Definition 6 Let S be some left-continuous t-conorm and T some t-norm with no zero divisors. Further, let µ be a weakly S-additive measure and B B 0. We say that µ T (. B) is µ-weakly S-additive, if the following is satisfied: S (µ T (A B), µ T (A 2 B)) = µ T ((A A 2 ) B) for A, A 2 B B such that µ(a A 2 ) µ(b). In the next example let Z = [0, ]. We will construct some T -conditional S-measures. Our condition will be B = [0, 0.5]: Example 2 (a) Take S M = max. Let µ be defined as follows: 0 if A = µ(a) = if A = Z 2 otherwise Then µ is weakly S M -additive. Then, regardless of the T -norm we consider, we get 0 if A B = µ T (A B) = otherwise I.e., µ T (. B) is S M -additive, but it is not a Lukasiewicz B B -filter. (b) Take the Lukasiewicz t-conorm S L and the minimum t-norm T M. Put µ equal to the Lebesgue measure on Z. Then 0 if A B = µ TM (A B) = if B A µ(a B) otherwise In this case µ TM (. B) is µ-weakly S L -additive and it is a Lukasiewicz B B -filter. (c) Take again the Lukasiewicz t-conorm S L and µ the Lebesgue measure on Z. Consider T H (the Hamacher product). Remind that T H (x, y) = x +, if min(x, y) > 0 y 0, if min(x, y) = 0. Then we get µ TH (A B) = 0 if A B = if B A µ(a B) µ(a B) otherwise I.e., µ TH (. B) is not µ-weakly S L -additive, but it is a Lukasiewicz B B -filter. 3. Minimum t-norm, T M, for A such that µ(b) = µ(a B) µ TM (A B) = µ(a B), otherwise Theorem 4 Let S be an arbitrary leftcontinuous t-conorm and µ some weakly S- additive measure. Then, for any B B 0, µ TM (. B) is µ-weakly S-additive. 3.2 Product t-norm, T P µ TP (A B) = µ(a B) µ(b) Theorem 5 Denote for a > 0 S a (x, y) = min, ( a x + a y) a}. Let S = S a for some a > 0 or S = S M and let µ be some weakly S-additive measure. Then, for any B B 0, µ TM (. B) is µ-weakly S-additive. Corollary (to Theorems 3 and 5) Let µ be some weakly S a -additive measure for a. Then, for any B B 0, µ TM (. B) is µ-weakly S-additive and it is a Lukasiewicz B B -filter. 3.3 Hamacher product t-norm, T H 0, if A = µ TH (A B) =, if A + µ(a B) µ(b) Theorem 6 Let µ be some weakly S M - additive measure. Then, for any B B 0, µ TH (. B) is µ-weakly S M -additive.

5 Example 3 Take again Z = [0, ] and B = [0, 0.5]. Choose some x [0, 0.5]. Then, µ : B [0, ] defined by µ(a) =, if A = Z 2, if x A Z 4, if x A 0, if A = is weakly S M -additive. The T H -conditional S M -measure is then the folowing, if x A µ TH (A B) = 0, if A = 3, if x A It is weakly S M -additive, but it is not a Lukasiewicz filter. On the other hand, in Example 2(c) we have constructed a T H -conditional S L -measure, which is not µ-weakly S L -additive, but which is a Lukasiewicz B B -filter. 4 Conclusion [3] Gutiérrez García, J., Mardones Pérez, I., Burton, M.H.(999), The relationship between various filter notions on a GLmonoid. J. Math. Anal. Appl., vol. 230, [4] Gutiérrez García, J., De Prada Vicente, M.A., Šostak, A.P. (2003), A unified approach to the concept of fuzzy L-uniform space, Rodabaugh, S.E. et al. (eds.) Topological and algebraic structures in fuzzy sets. Trends in Logic, Studia Logica Library, volume 20, Kluwer Acad. Publisher, Dordrecht, pages 8-4. [5] Kalina, M. (2005), Lukasiewicz filters and their Cartesian products. In: Proceedings of EUSFLAT 2005, Barcelona, Spain, pages [6] Pap, E. (995), Null-additive set functions, Ister Science, Bratislava, Kluwer Acad. Publishers, Dordrecht. In this paper we have shown a relationship between Lukasiewicz filters and the weak S- additivity of the corresponding measure µ. We have further introduced the T -conditional S-measures and studied (for some particular t-norms T and t-conorms S) if this conditional measure is µ-weakly S-additive or if it is a Lukasiewicz B-filter. Acknowledgements This work was supported by Science and Technology Assistance Agency under the contract No. APVT , and by the VEGA grant agency, grant number /304/06 References [] Burton, M.H., Muraleetharan, M., Gutiérrez García, J. (999), Generalised filters, Fuzzy Sets and Systems, vol. 06, pages [2] Burton, M.H., Muraleetharan, M., Gutiérrez García, J. (999), Generalised filters 2, Fuzzy Sets and Systems, vol. 06, pages